Question

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x} $$

Short answer

The limit is \( e \).

Step-by-step solution

Recognize the Form

The given limit is \( \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x \), which is a classic limit that resembles the definition of the number \( e \). Recognize that this expression can become indeterminate when analyzing the exponent as \( x \to \infty \).

Introduce Natural Logarithm

Introduce the natural logarithm to simplify the limit, letting \( y = \left(1 + \frac{1}{x}\right)^x \). Then take the natural logarithm of both sides, \( \ln y = x \ln\left(1 + \frac{1}{x}\right) \). The limit now becomes \( \lim_{x \to \infty} \ln y = \lim_{x \to \infty} x \ln\left(1 + \frac{1}{x}\right) \).

Simplify the Expression

Focus on the expression \( \ln\left(1 + \frac{1}{x}\right) \). For small values of \( \frac{1}{x} \), you can use the approximation \( \ln(1+u) \approx u \), so \( \ln\left(1 + \frac{1}{x}\right) \approx \frac{1}{x} \). Therefore, the expression simplifies to \( x \cdot \frac{1}{x} = 1 \).

Evaluate the Limit

Since \( \lim_{x \to \infty} x \ln\left(1 + \frac{1}{x}\right) = \lim_{x \to \infty} 1 = 1 \), it follows that \( \lim_{x \to \infty} \ln y = 1 \). Therefore, using the properties of logarithms, \( y = e^1 = e \).

Conclusion from Simplification

From the previous steps, conclude that since we let \( y = \left(1 + \frac{1}{x}\right)^x \) and found \( \ln y \to 1 \), the original limit \( \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e \).

Key concepts

Understanding Limits

In calculus, limits help us understand the behavior of functions as they approach a certain point or infinity. Limits are essential when dealing with functions that do not have a simple value at a specific point. Instead, they tell us what value the function moves towards as the input nears a particular number or direction.
In the exercise, we explore the limit \( \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x \). This means we are interested in what happens to the expression \( \left(1 + \frac{1}{x}\right)^x \) as \( x \) becomes very large. Limits are a foundational concept in calculus, paving the way for understanding continuity, derivatives, and integrals.

Indeterminate Forms and l'Hôpital's Rule

Sometimes, limits lead to expressions like \( 0/0 \), \( \infty/\infty \), or others that do not convey a clear outcome. These are called indeterminate forms. The specific form examined here is \( \left(1 + \frac{1}{x}\right)^x \) as \( x \to \infty \), which resembles a known indeterminate form that helps derive the mathematical constant \( e \).
Using l'Hôpital's Rule, a technique in calculus, we can address these indeterminate forms by differentiating the numerator and denominator separately, provided the limit approaches one of these indeterminate forms. However, for the exercise at hand, we address the indeterminate nature through logarithmic transformations instead.

The Role of Natural Logarithms

Natural logarithms simplify complex expressions involving exponential growth. For the given problem, we introduce the natural logarithm \( \ln \) to transform the expression.
By setting \( y = \left(1 + \frac{1}{x}\right)^x \), we take the natural logarithm to get \( \ln y = x \ln\left(1 + \frac{1}{x}\right) \). Transforming a power expression into a polynomial makes calculus operations more feasible.
The logarithm helps convert the complex power into a product, which is easier to handle, especially when taking limits or derivatives. Natural logs are integral in finding the limits of many functions, particularly those involving exponential forms.

The Importance of Number e

The number \( e \) is approximately equal to 2.71828 and is a fundamental constant in mathematics. It represents the base rate of growth shared by all continually growing processes.
The original expression \( \left(1 + \frac{1}{x}\right)^x \), as \( x \to \infty \), converges to \( e \). This expression is a classic calculus example used to define \( e \) through limits.
The importance of \( e \) extends beyond this exercise as it is pivotal in fields like finance, biology, and physics, representing natural growth in economics, population models, and radioactive decay. Its uniqueness arises from the fact that the derivative of \( e^x \) is \( e^x \) itself, showing it is fundamentally interlinked with the rate of growth.